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Line Integrals Examples 1

Recall from the Line Integrals page that if $z = f(x, y)$ is a two variable real-valued function and $C$ is a smooth plane curve defined by the parametric equations $x = x(t)$ and $y = y(t)$ then the line integral of $f$ along $C$ is given by:

Similarly if $z = f(x, y, z)$ is a three variable real-valued function and $C$ is a smooth space curve defined by the parametric equations $x = x(t)$, $y = y(t)$ and $z = z(t)$ then the line integral of $f$ along $C$ is given by:

Example 1

Evaluate the line integral $\int_C x - y \: ds$ where $C$ is the line segment from the point $(1, 3)$ to $(5, -2)$.

In order to evaluate this line integral, we will need to parameterize this line segment. We can parameterize this line segment as $\vec{r}(t) = (1 - t)(1, 3) + t(5, -2)$ which yields the parametric equations $x(t) = 1 + 4t$ and $y(t) = 3 - 5t$ for $0 ≤ t ≤ 1$. Therefore $\frac{dx}{dt} = 4$ and $\frac{dy}{dt} = -5$.

Example 2

Evaluate the line integral $\int_C y^3 \: ds$ where $C$ is the curve given by the parametric equations $x(t) = t^3$ and $y(t) = t$ for $0 ≤ t ≤ 2$.

We first note that $f(x, y) = y^3$ and so $f(x, y) = f(x(t), y(t)) = f(t^3, t) = t^3$. Also note that $\frac{dx}{dt} = 3t^2$ and $\frac{dy}{dt} = 1$. Applying the formula for evaluating line integrals for two variable functions and we get that: